The pigeonholeprinciple is an example of a counting argument which can be applied to many formal problems, including ones involving infinite sets that cannot be put into one-to-one correspondence.

If we assign a pigeonhole for each number of hairs on a head, and assign people to the pigeonhole with their number of hairs on it, there must be at least two people with the same number of hairs on their heads.

If we assign a pigeonhole for each number of hairs on a head, and assign people to the pigeonhole with their number of hairs on it, there must be two people with the same number of hairs on their heads.

The proof makes use of the PigeonholePrinciple: If n objects are distributed between fewer than n boxes, at least one box must contain at least two of the objects.

It follows from Dirichlet's box principle, that in any permutation of 10 distinct numbers there exists an increasing subsequence of at least 4 numbers or a decreasing subsequence of at least 4 numbers.

Dirichlet's box principle asserts that if n objects are put into m boxes, some box must contain at least ceil(n/m) objects, some box must contain at most floor(n/m).